1. Correlation, Energy Spectral Density and Power Spectral Density
Chapter 8

2. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Introduction
Relationships between signals can be just as important as characteristics of individual signals
The relationships between excitation and/or response signals in a system can indicate the nature of the system
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3. Flow Velocity Measurement
Heater
Thermometer
The relative timing of the two signals, p(t) and T(t),
and the distance, d, between the heater and thermometer
together determine the flow velocity.
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4. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Correlograms
Signals
Correlogram
Two Highly
Negatively
Correlated
DT Signals
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5. Correlograms Signals Correlogram Two Uncorrelated CT Signals 4/6/2019
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6. Correlograms Signals Correlogram Two Partially Correlated DT Signals
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7. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Correlograms
Signals
Correlogram
These two CT signals are not strongly correlated but would be if one were shifted in time the right amount
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8. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Correlograms
DT Sinusoids With
a Time Delay
CT Sinusoids With
a Time Delay
Signals
Signals
Correlogram
Correlogram
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9. Correlograms Signals Correlogram Two Non-Linearly Related DT Signals
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10. The Correlation Function
Positively Correlated DT Sinusoids with Zero Mean
Uncorrelated DT Sinusoids with
Zero Mean
Negatively Correlated DT Sinusoids with Zero Mean
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11. The Correlation Function
Positively Correlated Random CT Signals with Zero Mean
Uncorrelated Random CT Signals with
Zero Mean
Negatively Correlated Random CT Signals
with Zero Mean
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12. The Correlation Function
Positively Correlated CT Sinusoids with Non-zero Mean
Uncorrelated CT Sinusoids with
Non-zero Mean
Negatively Correlated CT Sinusoids with Non-zero Mean
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M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

13. The Correlation Function
Positively Correlated Random DT Signals with Non-zero Mean
Uncorrelated Random DT Signals with
Non-zero Mean
Negatively Correlated Random DT Signals
with Non-zero Mean
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14. Correlation of Energy Signals
The correlation between two energy signals, x and y, is the area under (for CT signals) or the sum of (for DT signals) the product of x and y.
The correlation function between two energy signals, x and y, is the area under (CT) or the sum of (DT) their product as a function of how much y is shifted relative to x. or or
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15. Correlation of Energy Signals
The correlation function for two energy signals is very similar
to the convolution of two energy signals.
Therefore it is possible to use convolution to find the correlation
function.
It also follows that or or or
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16. Correlation of Power Signals
The correlation function between two power signals, x and y, is the average value of their product as a function of how much y is shifted relative to x.
If the two signals are both periodic and their fundamental periods have a finite least common period,
where T or N is any integer multiple of that least common period. or or
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17. Correlation of Power Signals
Correlation of periodic signals is very similar to periodic
convolution
where it is understood that the period of the periodic convolution is any integer multiple of the least common period of the two fundamental periods of x and y. or or
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18. Correlation of Power Signals
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19. Correlation of Sinusoids
The correlation function for two sinusoids of different frequencies is always zero. (pp )
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20. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Autocorrelation
A very important special case of correlation is autocorrelation. Autocorrelation is the correlation of a function with a shifted version of itself. For energy signals,
At a shift, t or m, of zero,
which is the signal energy of the signal. For power signals,
which is the average signal power of the signal. or or or
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21. Properties of Autocorrelation
Autocorrelation is an even function.
Autocorrelation magnitude can never be larger than it is at zero shift.
If a signal is time shifted its autocorrelation does not change.
The autocorrelation of a sum of sinusoids of different frequencies is the sum of the autocorrelations of the individual sinusoids. or or
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22. Autocorrelation Examples
Three different random DT signals and their autocorrelations. Notice that, even though the signals are different, their autocorrelations are quite similar, all peaking sharply at a shift of zero.
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23. Autocorrelation Examples
Autocorrelations for a cosine “burst” and a sine “burst”. Notice that they are almost (but not quite) identical.
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24. Autocorrelation Examples
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Matched Filters
A very useful technique for detecting the presence of a signal of a certain shape in the presence of noise is the matched filter.
The matched filter uses correlation to detect the signal so this filter is sometimes called a correlation filter
It is often used to detect 1’s and 0’s in a binary data stream
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Matched Filters
It has been shown that the optimal filter to detect a noisy signal is one whose impulse response is proportional to the time inverse of the signal. Here are some examples of waveshapes encoding 1’s and 0’s and the impulse responses of matched filters.
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Matched Filters
Noiseless Bits
Noisy Bits
Even in the presence of a large additive noise signal the matched filter indicates with a high response level the presence of a 1 and with a low response level the presence of a 0. Since the 1 and 0 are encoded as the negatives of each other, one matched filter optimally detects both.
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28. Autocorrelation Examples
Two familiar DT signal shapes and their autocorrelations.
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29. Autocorrelation Examples
Three random power signals with different frequency content
and their autocorrelations.
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30. Autocorrelation Examples
Autocorrelation functions for a cosine and a sine. Notice
that the autocorrelation functions are identical even though
the signals are different.
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31. Autocorrelation Examples
One way to simulate a random signal is with a summation of sinusoids of different frequencies and random phases
Since all the sinusoids have different frequencies the autocorrelation of the sum is simply the sum of the autocorrelations
Also, since a time shift (phase shift) does not affect the autocorrelation, when the phases are randomized the signals change, but not their autocorrelations
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32. Autocorrelation Examples
Let a random signal be described by
Since all the sinusoids are at different frequencies,
where is the autocorrelation of
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33. Autocorrelation Examples
Four Different
Random Signals
with Identical
Autocorrelations
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34. Autocorrelation Examples
Four Different
Random Signals
with Identical
Autocorrelations
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35. Autocorrelation Examples
Four Different
Random Signals
with Identical
Autocorrelations
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36. Autocorrelation Examples
Four Different
Random Signals
with Identical
Autocorrelations
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Cross Correlation
Cross correlation is really just “correlation” in the cases in
which the two signals being compared are different. The name is commonly used to distinguish it from autocorrelation.
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Cross Correlation
A comparison of x and y with y shifted for maximum
correlation.
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39. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Cross Correlation
Below, x and z are highly positively correlated and x and y are uncorrelated. All three signals have the same average signal power. The signal power of x+z is greater than the signal power of x+y.
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40. Correlation and the Fourier Series
Calculating Fourier series harmonic functions can be thought of as a process of correlation. Let
Then the trigonometric CTFS harmonic functions are
Also, let
then the complex CTFS harmonic function is
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41. Energy Spectral Density
The total signal energy in an energy signal is
The quantity, , or , is called the energy spectral density (ESD) of the signal, x and is conventionally given the symbol, Y. That is,
It can be shown that if x is a real-valued signal that the ESD is even, non-negative and real. or or
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42. Energy Spectral Density
Probably the most important fact about ESD is the relationship between the ESD of the excitation of an LTI system and the ESD of the response of the system. It can be shown (pp ) that they are related by or
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43. Energy Spectral Density
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44. Energy Spectral Density
It can be shown (pp ) that, for an energy signal, ESD and autocorrelation form a Fourier transform pair. or
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45. Power Spectral Density
Power spectral density (PSD) applies to power signals in the same way that energy spectral density applies to energy signals. The PSD of a signal x is conventionally indicated by the notation, or In an LTI system,
Also, for a power signal, PSD and autocorrelation form a Fourier transform pair. or or
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46. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
PSD Concept
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47. Typical Signals in PSD Concept
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